The Mukai pairing—II: the Hochschild–Kostant–Rosenberg isomorphism

Abstract

We continue the study of the Hochschild structure of a smooth space that we began in our previous paper, examining implications of the Hochschild–Kostant–Rosenberg theorem. The main contributions of the present paper are: • we introduce a generalization of the usual notions of Mukai vector and Mukai pairing on differential forms that applies to arbitrary manifolds; • we give a proof of the fact that the natural Chern character map K 0 ( X ) → HH 0 ( X ) becomes, after the HKR isomorphism, the usual one K 0 ( X ) → ⊕ H i ( X , Ω X i ) ; and • we present a conjecture that relates the Hochschild and harmonic structures of a smooth space, similar in spirit to the Tsygan formality conjecture.